The wedgie: Difference between revisions
Wikispaces>genewardsmith **Imported revision 290312289 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 290417595 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-08 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-08 17:52:22 UTC</tt>.<br> | ||
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If C = ||W|| is the TE complexity, then the formula for the [[Tenney-Euclidean metrics#Logflat TE badness|logflat badness]] B in the 7-limit rank-two case is particularly simple: B = CE. If complexity is bounded by, for example, 20 (which allows for some quite complex temperaments) then since E ≤ 1/(4√q5√q7), B ≤ 20/(4√q5√q7) = 240.250. This is an absurdly high badness figure; while simply bounding complexity will lead to a finite list, the list would be enormous. An alternative is also to bound badness; for instance, we might produce a list of 7-limit rank-two temperaments with complexity less than 20 and a more reasonable badness limit, such as 0.05 or 0.06. | If C = ||W|| is the TE complexity, then the formula for the [[Tenney-Euclidean metrics#Logflat TE badness|logflat badness]] B in the 7-limit rank-two case is particularly simple: B = CE. If complexity is bounded by, for example, 20 (which allows for some quite complex temperaments) then since E ≤ 1/(4√q5√q7), B ≤ 20/(4√q5√q7) = 240.250. This is an absurdly high badness figure; while simply bounding complexity will lead to a finite list, the list would be enormous. An alternative is also to bound badness; for instance, we might produce a list of 7-limit rank-two temperaments with complexity less than 20 and a more reasonable badness limit, such as 0.05 or 0.06. | ||
Essentially the same situation obtains for rank two temperaments in higher limits. The rule then is that if E ≤ 1/(C(n, 3)√lb(q)√lb(p)) then wedging <1 lb(3) lb(5) ... lb(p)| with the val consisting of 0 followed by the first n-1 coefficients of the wedgie and rounding will give the wedgie. Here p and q are the largest and second largest primes in the prime limit, lb(x) is log base two, and C(n, 3) is n choose three, n(n-1)(n-2)/6. | Essentially the same situation obtains for rank two temperaments in higher limits. The rule then is that if E ≤ 1/(C(n, 3)√lb(q)√lb(p)) then wedging K = <1 lb(3) lb(5) ... lb(p)| with the val consisting of 0 followed by the first n-1 coefficients of the wedgie and rounding will give the wedgie. Here p and q are the largest and second largest primes in the prime limit, lb(x) is log base two, and C(n, 3) is n choose three, n(n-1)(n-2)/6. | ||
In general, we can reconstruct W by rounding (W∨2)∧K to the nearest integer coefficients, where K is the JI point <1 lb(3) lb(5) ... lb(p)| in unweighted coordinates. Since (W-(W∨2)∧K)∧K = W∧K, we have that relative error, ||W∧K||, equals ||Y∧K|| where Y = (W∨2)∧K. Hence relative error for W can be bounded in terms of the coefficients of W∨2. Moreover, ||W|| = ||(W-Y)+Y|| ≤ ||W-Y|| + ||Y|| by the triangle inequality, and since ||W-Y|| is bounded by the fact that W has been obtained by rounding, complexity, which is ||W||, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we can be conducting a search. | |||
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If C = ||W|| is the TE complexity, then the formula for the <a class="wiki_link" href="/Tenney-Euclidean%20metrics#Logflat TE badness">logflat badness</a> B in the 7-limit rank-two case is particularly simple: B = CE. If complexity is bounded by, for example, 20 (which allows for some quite complex temperaments) then since E ≤ 1/(4√q5√q7), B ≤ 20/(4√q5√q7) = 240.250. This is an absurdly high badness figure; while simply bounding complexity will lead to a finite list, the list would be enormous. An alternative is also to bound badness; for instance, we might produce a list of 7-limit rank-two temperaments with complexity less than 20 and a more reasonable badness limit, such as 0.05 or 0.06.<br /> | If C = ||W|| is the TE complexity, then the formula for the <a class="wiki_link" href="/Tenney-Euclidean%20metrics#Logflat TE badness">logflat badness</a> B in the 7-limit rank-two case is particularly simple: B = CE. If complexity is bounded by, for example, 20 (which allows for some quite complex temperaments) then since E ≤ 1/(4√q5√q7), B ≤ 20/(4√q5√q7) = 240.250. This is an absurdly high badness figure; while simply bounding complexity will lead to a finite list, the list would be enormous. An alternative is also to bound badness; for instance, we might produce a list of 7-limit rank-two temperaments with complexity less than 20 and a more reasonable badness limit, such as 0.05 or 0.06.<br /> | ||
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Essentially the same situation obtains for rank two temperaments in higher limits. The rule then is that if E ≤ 1/(C(n, 3)√lb(q)√lb(p)) then wedging &lt;1 lb(3) lb(5) ... lb(p)| with the val consisting of 0 followed by the first n-1 coefficients of the wedgie and rounding will give the wedgie. Here p and q are the largest and second largest primes in the prime limit, lb(x) is log base two, and C(n, 3) is n choose three, n(n-1)(n-2)/6.</body></html></pre></div> | Essentially the same situation obtains for rank two temperaments in higher limits. The rule then is that if E ≤ 1/(C(n, 3)√lb(q)√lb(p)) then wedging K = &lt;1 lb(3) lb(5) ... lb(p)| with the val consisting of 0 followed by the first n-1 coefficients of the wedgie and rounding will give the wedgie. Here p and q are the largest and second largest primes in the prime limit, lb(x) is log base two, and C(n, 3) is n choose three, n(n-1)(n-2)/6.<br /> | ||
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In general, we can reconstruct W by rounding (W∨2)∧K to the nearest integer coefficients, where K is the JI point &lt;1 lb(3) lb(5) ... lb(p)| in unweighted coordinates. Since (W-(W∨2)∧K)∧K = W∧K, we have that relative error, ||W∧K||, equals ||Y∧K|| where Y = (W∨2)∧K. Hence relative error for W can be bounded in terms of the coefficients of W∨2. Moreover, ||W|| = ||(W-Y)+Y|| ≤ ||W-Y|| + ||Y|| by the triangle inequality, and since ||W-Y|| is bounded by the fact that W has been obtained by rounding, complexity, which is ||W||, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we can be conducting a search.</body></html></pre></div> | |||